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Mirrors > Home > MPE Home > Th. List > sbcieg | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
sbcieg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbcieg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | sbcieg.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | sbciegf 3806 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 df-sbc 3770 |
This theorem is referenced by: sbcie 3809 2nreu 4389 ralsngOLD 4600 rexsngOLD 4601 reuprg0 4630 rabsnif 4651 ralrnmptw 6852 ralrnmpt 6854 fpwwe2lem3 10043 nn1suc 11647 opfi1uzind 13847 mndind 17980 fgcl 22414 cfinfil 22429 csdfil 22430 supfil 22431 fin1aufil 22468 ifeqeqx 30224 nn0min 30463 bnj1452 32221 cdlemk35s 37953 cdlemk39s 37955 cdlemk42 37957 2nn0ind 39420 zindbi 39421 prproropreud 43548 |
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